Unveiling the S=3/2 Kitaev honeycomb spin liquids

The S=3/2 Kitaev honeycomb model (KHM) is a quantum spin liquid (QSL) state coupled to a static Z2 gauge field. Employing an SO(6) Majorana representation of spin3/2’s, we find an exact representation of the conserved plaquette fluxes in terms of static Z2 gauge fields akin to the S=1/2 KHM which enables us to treat the remaining interacting matter fermion sector in a parton mean-field theory. We uncover a ground-state phase diagram consisting of gapped and gapless QSLs. Our parton description is in quantitative agreement with numerical simulations, and is furthermore corroborated by the addition of a [001] single ion anisotropy (SIA) which continuously connects the gapless Dirac QSL of our model with that of the S=1/2 KHM. In the presence of a weak [111] SIA, we discuss an emergent chiral QSL within a perturbation theory.


June 14, 2022
In this Supplementary Information, we show more details about (i) the self-consistent meanfield theory for S=3/2 Kitaev honeycomb model and (ii) the spin quadrupolar parameter.

Supplementary Note 1: Self-consistent mean-field theory
In the main text, we have introduced a mean-field Hamiltonian H({u} = 1) with the following mean-field order parameters: For concreteness, we investigated the mean-field theory of the model with fixed exchange parameters J x = J y = 1 and varying J z and D z . In this parameter regime, the model preserves mirror symmetry M z , inversion symmetry I, and time-reversal symmetry T (see Supplementary Fig. 1), which will impose constraints to the order parameters and allow us to provide a succinct form for the mean-field Hamiltonian H({u} = 1). In accordance with a full symmetry analysis [1], we find that for a translational invariant solution which preserves M z , I, and T symmetries there exist only eight non-zero and independent mean-field order parameters where the bond parameters ∆ ab x(z) are defined on the x-type (z-type) bonds. We can write down the mean-field Hamiltonian H MF ({u} = 1) in the reciprocal space as where the Fourier transformed fermions on the A (B) sublattice are complex fermions rather than Majorana fermions, and r denotes the Here J x = J y = 1 and a zero-flux configuration of {u} = 1 is chosen. Notice that the selfconsistent solution for effective isotropic S=1/2 Kitaev spin liquid at (J z = 4, D z → ∞) is equivalent to that given in Ref. [3] up to a factor of −2, where the minus sign is caused by the antiferromagnetic couplings J a > 0 and the factor of 2 is caused by the normalization condition of Majorana fermions used here, e.g., (θ a i ) 2 = (η a i ) 2 = 1.
unit cell coordinates. Then, the self-consistent equations can be written as

Supplementary Note 2: Spin quadrupolar parameter
The spin quadrupolar parameter, which is distinguished from magnetic order, is time-reversal invariant. This order usually does not exist in the S = 1/2 systems because a product of arbitrary two spin-1/2 operators is still a spin-1/2 operator or a trivial identity matrix. While for higher spin systems, the product of two spin operators gives rise to nontrivial spin quadrupolar operators which generally can support the spin quadrupolar parameters. A simple example of spin quadrupolar parameter for S = 1 systems can be found in Ref. [2].
The Z 2 quantum spin liquid state |Ψ shown in the phase diagram in the main text generally coexists with a spin quadrupolar parameter, e.g.,  The spin quadrupolar parameter usually is illustrated by the probabilities of spin fluctua- . Since |Ψ is a many-body state, for illustration purposes we utilize the probabilities of spin fluctuations between S=3/2 spin coherent states |S(n) j and the reduced density matrix ρ j (Q xy ) for site j: where |S j (n) with n · S j |S j (n) = S|S j (n) n is a spin coherent state pointing to the direction ofn. The probabilities of spin fluctuations for Q xy = −1 and Q xy = +1 are shown in Supplementary Figs. 3(a) and (b), respectively. Here the reduced density matrix ρ j (Q xy ), which can resolve Eq. (4), is obtained by exact diagonalization on a 2 × 2 torus.